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The singularvalue decomposition theorem says that M has a factorization of the form
In any such singular value decomposition, the diagonal entries of Σ are necessarily equal to the singular values of M.
The columns u_{1},...,u_{m} of U are eigenvectors of MM^{*} and are left singular vectors of M. The columns v_{1},...,v_{n} of V are eigenvectors of M^{*}M and are right singular vectors of M. Note however that different singular value decompositions of M can contain different singular vectors.
The linear transformation T: K^{n} → K^{m} that takes a vector x to Mx has a particularly simple description with respect to these orthonormal bases: we have T(v_{i}) = d_{i} u_{i}, for i = 1,...,min(m,n), where d_{i} is the ith diagonal entry of D, and T(v_{i}) = 0 for i > min(m,n).
The number of nonzero singular values is equal to the rank r of M. These nonzero singular values are equal to the square roots of the nonzero eigenvalues of the positive semidefinite[?] matrix MM^{*}, and also equal to the square roots of the nonzero eigenvalues of M^{*}M.
If we focus only on these r nonzero singular values, we can construct a singularvalue decomposition of the following type:
where G is an mbyr orthonormal matrix over K, H is an nbyr orthonormal matrix over K and D is an rbyr diagonal matrix whose diagonal entries are positive real numbers.
The sum of the k largest singular values of M is a matrix norm, the Ky Fan knorm of M. The Ky Fan 1norm is just the operator norm[?] of M as a linear operator with respect to the Euclidean norms of K^{m} and K^{n}.
See also: matrix decomposition
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